Trust-region subproblem (TRS) is an important problem arising in many applications such as numerical optimization, Tikhonov regularization of ill-posed problems, and constrained eigenvalue problems. In recent decades, extensive works focus on how to solve the trust-region subproblem efficiently. To the best of our knowledge, there are few results on perturbation analysis of the trust-region subproblem. In order to fill in this gap, we focus on first-order perturbation theory of the trust-region subproblem. The main contributions of this paper are three-fold. First, suppose that the TRS is in easy case, we give a sufficient condition under which the perturbed TRS is still in easy case. Second, with the help of the structure of the TRS and the classical eigenproblem perturbation theory, we perform first-order perturbation analysis on the Lagrange multiplier and the solution of the TRS, and define the condition numbers of them. Third, we point out that the solution and the Lagrange multiplier could be well-conditioned even if TRS is in nearly hard case. The established results are computable, and are helpful to evaluate ill-conditioning of the TRS problem beforehand. Numerical experiments show the sharpness of the established bounds and the effectiveness of the proposed strategies.
翻译:信任区域的子问题(TRS)是许多应用中出现的一个重要问题,例如数字优化、对错误问题的蒂克诺诺夫的正规化和受限制的二元价值问题。近几十年来,大量工作的重点是如何有效解决信任区域的子问题。据我们所知,对信任区域的子问题进行扰动分析的结果很少。为了填补这一空白,我们侧重于信任区域子问题的第一阶扰动理论。本文的主要贡献是三重。首先,假设TRS是容易的,我们给出了充分的条件,使被包围的TRS仍然很容易解决。第二,根据TRS的结构和典型的易碎裂理论,我们进行关于拉格兰奇的乘数和拟议的TRS解决办法的第一阶扰动分析,并界定了它们的条件。第三,我们指出,解决方案和拉格兰基的乘数是三,假设TRS是容易发生的,即使TRS的精确度是稳定的试验结果,也能够很好地表明,即使TRS的精确度是稳定的试验,也几乎是困难的。